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@@ -496,7 +496,7 @@ corresponding modulus. \section{Comparison to experiment} \Rus\ experiments~\cite{ghosh_single-component_nodate} yield the individual -elastic moduli broken into irrep symmetries; data for the $\Bog$ and $\Btg$ +elastic moduli broken into irreps; data for the $\Bog$ and $\Btg$ components defined in \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b). The $\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't appear to have any response to the presence of the transition, exhibiting the @@ -546,44 +546,36 @@ pressure, where the depth of the cusp in the $\Bog$ modulus should deepen (perhaps with these commensurability jumps) at low pressure and approach zero as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. Alternatively, \rus\ done at ambient pressure might examine the heavy Fermi -liquid to \afm\ transition by doping. Though previous \rus\ studies have doped -\urusi\ with Rhodium,\cite{yanagisawa_ultrasonic_2014} the magnetic rhodium dopants likely promote magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully explore -the transition out of the HO phase. Our work also motivates experiments +liquid to \afm\ transition by doping. Our work also motivates experiments that can probe the entire correlation function---like x-ray and neutron scattering---and directly resolve its finite-$q$ divergence. The presence of spatial commensurability is known to be irrelevant to critical behavior at a one-component disordered to modulated transition, and therefore is not -expected to modify the thermodynamic behavior -otherwise.\cite{garel_commensurability_1976} +expected to otherwise modify the thermodynamic behavior.\cite{garel_commensurability_1976} There are two apparent discrepancies between the orthorhombic strain in the -phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018} -and that predicted by our mean field theory when its uniform ordered phase is +phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018}, +and that predicted by our mean field theory if its uniform $\Bog$ phase is taken to be coincident with \urusi's \afm. The first is the apparent onset of -the orthorhombic phase in the \ho\ state prior to the onset of \afm. As the -recent x-ray research\cite{choi_pressure-induced_2018} notes, this could be due +the orthorhombic phase in the \ho\ state at slightly lower pressures than the onset of \afm. As the +recent x-ray research\cite{choi_pressure-induced_2018} notes, this misalignment of the two transitions as function of doping could be due to the lack of an ambient pressure calibration for the lattice constant. The second discrepancy is the onset of orthorhombicity at higher temperatures than -the onset of \afm. Susceptibility data sees no trace of another phase -transition at these higher temperatures.\cite{inoue_high-field_2001} We suspect -that the high-temperature orthorhombic signature is not the result of a bulk -phase, and could be due to the high energy (small-wavelength) nature of x-rays -as an experimental probe: \op\ fluctuations should lead to the formation of -orthorhombic regions on the order of the correlation length that become larger -and more persistent as the transition is approached. +the onset of \afm. We note that magnetic susceptibility data sees no trace of another phase +transition at these higher temperatures. \cite{inoue_high-field_2001} It is therefore possible that the high-temperature orthorhombic signature in x-ray scattering is not the result of a bulk thermodynamic phase, but instead marks the onset of short-range correlations, as it does in the high-T$_{\mathrm{c}}$ cuprates \cite{ghiringhelli2012long} (where the onset of CDW correlations also lacks a thermodynamic phase transition). Three dimensions is below the upper critical dimension $4\frac12$ of a -one-component disordered to modulated transition, and so mean field theory +one-component disordered-to-modulated transition, and so mean field theory should break down sufficiently close to the critical point due to fluctuations, at the Ginzburg temperature. \cite{hornreich_lifshitz_1980, -ginzburg_remarks_1961} Magnetic phase transitions tend to have Ginzburg +ginzburg_remarks_1961} Magnetic phase transitions tend to have a Ginzburg temperature of order one. Our fit above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about what one would expect for a generic magnetic transition. The agreement of this data in the $t\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see -deviations from mean field behavior within around several degrees Kelvin of the +deviations from mean field behavior within approximately several Kelvin of the critical point. An ultrasound experiment with more precise temperature resolution near the critical point may be able to resolve a modified cusp exponent $\gamma\simeq1.31$,\cite{guida_critical_1998} since the universality @@ -613,10 +605,10 @@ uniform $\Bog$ electronic order. The corresponding prediction of uniform $\Bog$ symmetry breaking in the high pressure phase is consistent with recent diffraction experiments, \cite{choi_pressure-induced_2018} except for the apparent earlier onset in -temperature of the $\Bog$ symmetry breaking, which we believe to be due to -fluctuating order above the actual phase transition. This work motivates both +temperature of the $\Bog$ symmetry breaking, which we believe may be due to +fluctuating order at temperatures above the actual transition temperature. This work motivates both further theoretical work regarding a microscopic theory with modulated $\Bog$ -order, and preforming \rus\ experiments at pressure that could further support +order, and preforming symmetry-sensitive thermodynamic experiments at pressure, such as ultrasound, that could further support or falsify this idea. \begin{acknowledgements} |